Large Density Ratio Research Articles

A consistent and conservative Phase-Field method for multiphase incompressible flows

In the present study, a consistent and conservative Phase-Field method, including both the model and scheme, is developed for multiphase flows with an arbitrary number of immiscible and incompressible fluid phases. The consistency of mass conservation and the consistency of mass and momentum transport are implemented to address the issue of physically coupling the Phase-Field equation, which locates different phases, to the hydrodynamics. These two consistency conditions, as illustrated, provide the “optimal” coupling because (i) the new momentum equation resulting from them is Galilean invariant and implies the kinetic energy conservation, regardless of the details of the Phase-Field equation, and (ii) failures of satisfying the second law of thermodynamics or the consistency of reduction of the multiphase flow model only result from the same failures of the Phase-Field equation but are not due to the new momentum equation. Physical interpretation of the consistency conditions and their formulations are first provided, and general formulations that are obtained from the consistency conditions and independent of the interpretation of the velocity are summarized. Then, the present consistent and conservative multiphase flow model is completed by selecting a reduction consistent Phase-Field equation. Several novel techniques are developed to inherit the physical properties of the multiphase flows after discretization, including the gradient-based phase selection procedure, the momentum conservative method for the surface force, and the general theorems to preserve the consistency conditions on the discrete level. Equipped with those novel techniques, a consistent and conservative scheme for the present multiphase flow model is developed and analyzed. The scheme satisfies the consistency conditions, conserves the mass and momentum, and assures the summation of the volume fractions to be unity, on the fully discrete level and for an arbitrary number of phases. All those properties are numerically validated. Numerical applications demonstrate that the present model and scheme are robust and effective in studying complicated multiphase dynamics, especially for those with large-density ratios.

A weakly compressible smoothed particle hydrodynamics framework for melting multiphase flow

In this study, the transient process of solid–liquid phase change is modeled and simulated by the multiphase smoothed particle hydrodynamics (SPH) method. First, to simulate the interfacial behaviors of melt liquids, the multiphase SPH model is established for immiscible viscous fluids with a large density ratio, where the environmental liquid surrounding the solid phase is considered, and the surface tension of the melt liquid can be accurately modeled by the continuum surface force method. Based on the multiphase model, the thermal dynamics model is incorporated to describe the heat conduction process. The solid–liquid phase change is realized by directly switching the state of the concerned SPH particle, where the absorbed latent heat is computed by the phase change model. Second, the model is validated by several simulation cases, including the Stefan problem, hydrostatic pressure of the evolving fluid interface, rising of two bubbles, and square droplet deformation, and the effects of numerical parameters on simulation accuracy and stability are also discussed. Third, the integrated SPH model is applied to simulate molten droplet formation and dropping processes. The results show that an initial solid–liquid interface disappears during the melting process, and new liquid–liquid interfaces gradually form and evolve under the action of surface tension, gravity, and viscosity. Phenomena such as thin-layer fluid dynamics and capillary instabilities are also reproduced, showing the effectiveness of the model for handling multiphase flow with heat conduction and phase change.

Development of a Compact Multivariable Sensor Probe for Two-Phase Detection in High Temperature PbLi–argon Vertical Columns

Liquid gas two phase flow is a common occurrence in various industrial applications. For nuclear fusion applications with a Li based breeder, existence of two phase flow may lead to critical issues including decreased tritium breeding ratio, generation of hot spots and improper nuclear shielding. Additionally, a very large density ratio of liquid metal to gas mandates relevant experiments towards development and validation of software tools. PbLi has gained immense focus for its various advantages and is utilized in several breeding blanket concepts. In view of above mentioned requirements, a two phase detection tool is imperative for liquid PbLi environment. For liquid metal applications, electrical conductivity based probes are most suitable in terms of ruggedness, fabrication ease and operational simplicity. However, corrosive nature and high operational temperature for PbLi put severe demands on electrical insulation, a foremost requirement for electrical conductivity based schemes. In this study, a multivariable probe based on electrical conductivity and temperature measurement schemes is developed using Al2O3 coating as electrical insulation. Developed probe is validated in PbLi/Ar vertical column with bulk PbLi temperature upto 400 degC and time averaged void fraction upto 0.95 covering flow regimes from dispersed bubbly flow upto in box loss of coolant accident. Developed probe provides high reliability and excellent temporal resolution towards individual bubble detection through electrical conductivity based principle alongwith simultaneous two phase temperature trends. Present paper provides details about sensor probe fabrication, calibration, two phase test facility, void fraction estimations, bubble frequency and bubble residence time estimations alongwith critical observations from preliminary experimental investigations.

Fully compressible multiphase model for computation of compressible fluid flows with large density ratio and the presence of shock waves

We propose a fully compressible multiphase model for simulating compressible interfacial flows. The mathematical model is formulated on the basis of the typical conservation laws for mixtures, including mass, momentum, and energy balance. The mass balance equation in each phase is considered to discretely conserve the mass of each phase; therefore, it is always mass conservative for each phase. The numerical solution procedure for this model is developed using a high-resolution shock-capturing finite-volume method based on the Harten–Lax–van Leer-contact (HLLC) Riemann solver and total variation diminishing (TVD) time-integration schemes for simulating unsteady multiphase flows and capturing strong shocks. The finite-volume Riemann solver is combined with a monotonic upstream-centered scheme for conservation laws (MUSCL) and compressive limiters to maintain a sharp interface. The numerical framework yields a conceptually simple but robust and versatile numerical procedure. Numerical results of benchmark Riemann problems demonstrate the ability of the proposed method to obtain sharp interfaces, free of spurious oscillations, high accuracy, conservation properties, and thermodynamic consistency over various Mach numbers.

Improved phase-field-based lattice Boltzmann method for thermocapillary flow.

In this paper, we present an improved phase-field-based lattice Boltzmann (LB) method for thermocapillary flows with large density, viscosity, and thermal conductivity ratios. The present method uses three LB models to solve the conservative Allen-Cahn equation, the incompressible Navier-Stokes equations, and the temperature equation. To overcome the difficulty caused by the convection term in solving the convection-diffusion equationfor the temperature field, we first rewrite the temperature equationas a diffuse equationwhere the convection term is regarded as the source term and then construct an improved LB model for the diffusion equation. The macroscopic governing equationscan be recovered correctly from the present LB method; moreover, the present LB method is much simpler and more efficient. In order to test the accuracy of this LB method, several numerical examples are considered, including the planar thermal Poiseuille flow of two immiscible fluids, the two-phase thermocapillary flow in a nonuniformly heated channel, and the thermocapillary Marangoni flow of a deformable bubble. It is found that the numerical results obtained from the present LB method are consistent with the theoretical prediction and available numerical data, which indicates that the present LB method is an effective approach for the thermocapillary flows.

A comparative study of the cumulant lattice Boltzmann method in a single-phase free-surface model of violent flows

Many coastal and ocean engineering flows, such as tsunamis inundating urban areas, tend to be violent. The characteristics of these damaging flow fields are three-dimensional, highly non-linear and non-hydrostatic. A fully three-dimensional free-surface fluid model is required to simulate such a flow field. Fluid simulations in the field of coastal engineering are often large-scale since large areas are the subject of the simulations. The numerical model must be not only accurate but also efficient. In recent years, the lattice Boltzmann method (LBM) has attracted much attention as a novel simulation method and has been successfully applied to various engineering fields. The LBM calculates complex phenomena in a simple framework. The density field determines the pressure field with the equation of state. This means that the LBM does not have to solve the pressure Poisson equation. However, the existing gas–liquid multi-phase flow models that employ the LBM have a critical problem. Pressure calculations with large density ratios easily become unstable, and the application of these models to violent flow fields is limited. The single-phase free-surface fluid model provides good approximations for flow problems in which the gas dynamics can be neglected. Moreover, the cumulant LBM has attracted attention because it has excellent numerical stability even for high Reynolds number flows. The single-phase free-surface flow model using the cumulant LBM is a suitable approach for simulating violent flow fields in coastal engineering. In this study, we propose a single-phase free-surface flow model based on the cumulant LBM using the volume-of-fluid (VOF) model to capture the interface. We demonstrate that the cumulant LBM is stable under violent flows and reproduces the density field well compared with the traditional single relaxation time model. We find that a larger bulk viscosity can reduce the numerical oscillation of the impact pressure acting on a structure, although a bulk viscosity that is too large reduces the accuracy and stability. The results of the proposed model are in good agreement with previous experimental results.

A generalized conservative phase-field simplified lattice Boltzmann method for miscible and immiscible ternary flows with large density ratio

In this paper, a generalized conservative phase-field simplified lattice Boltzmann method is proposed, which is suitable for both miscible and immiscible ternary flow problems. This method extends our earlier simplified multiphase lattice Boltzmann method (SMLBM) (Chen et al., 2018a) for two-phase flows to the ternary flows by using a generalized conservative equation with Lagrange multiplier to control the evolution of the interface and ensures the conservations of the volume and total mass of each phase. Moreover, the good stability of the SMLBM is utilized for solving the interface problems with large gradients induced by large density ratios between different fluid components. To validate the present method, several ternary flow examples are simulated. The numerical results show that the method can effectively simulate the ternary flows with a density ratio up to 1200 and can be applied to accurately simulate the miscible ternary flows with satisfying the reduction consistency conditions.

A lattice Boltzmann formulation of the one-fluid model for multiphase flow

A multiphase lattice Boltzmann model is constructed to numerically solve the one-fluid flow equations for immiscible fluids. The method features one solver for the macroscopic pressure and momentum and another for a scalar field that captures and sharpens the interface. The surface tension is set a priori and independently of other parameters. The interface capillary tensor is embedded within the moments of the lattice Boltzmann equation so that its divergence is captured locally. The algorithm is simple and can compute flows with large density and viscosity ratios while maintaining distributed but narrow interfaces. The model is validated against analytical solutions and benchmark simulations.

Phase-field lattice Boltzmann model for two-phase flows with large density ratio.

In this work, a lattice Boltzmann (LB) model based on the phase-field method is proposed for simulating large density ratio two-phase flows. An improved multiple-relaxation-time (MRT) LB equation is first developed to solve the conserved Allen-Cahn (AC) equation. By utilizing a nondiagonal relaxation matrix and modifying the equilibrium distribution function and discrete source term, the conserved AC equation can be correctly recovered by the proposed MRT LB equation with no deviation term. Therefore, the calculations of the temporal derivative term in the previous LB models are successfully avoided. Numerical tests demonstrate that satisfactory accuracy can be achieved by the present model to solve the conserved AC equation. What is more, the discrete force term of the MRT LB equation for the incompressible Navier-Stokes equations is also simplified and modified in the present work. An alternative scheme to calculate the gradient terms of the order parameter involved in the discrete force term through the nonequilibrium part of the distribution function is also developed. To validate the ability of the present LB model for simulating large density ratio two-phase flows, series of benchmarks, including two-phase Poiseuille flow, droplet impacting on thin liquid film, and planar Taylor bubble are simulated. It is found that the results predicted by the present LB model agree well with the analytical, numerical, and experimental results.

An adaptive mesh refinement‐multiphase lattice Boltzmann flux solver for three‐dimensional simulation of droplet collision

AbstractA three‐dimensional adaptive mesh refinement‐multiphase lattice Boltzmann flux solver (3D‐AMR‐MLBFS) is developed for effectively simulating complex multiphase flows with large density ratio and high Reynolds number. In the method, the multiphase lattice Boltzmann flux solver (MLBFS) is used to solve the flow field and the level set method is adopted to capture the interface. In addition, a free energy‐based continuum surface tension force (FE‐CSF) model, which has a clear physical basis and does not need to introduce a regularization function, is applied to calculate the surface tension force. In order to improve the computational efficiency and at the same time ensure accuracy, the adaptive mesh refinement technology is introduced. This method is used to simulate three‐dimensional collision of two equal‐sized droplets for a wide range of parametric conditions. We not only verify the developed method by comparing with available literature data but also successfully reproduce four typical outcomes of droplet collision, namely coalescence, reflexive separation, stretching separation, and liquid membrane shattering, for which the evolutions of droplet morphologies and various energies are revealed.

A stable SPH model with large CFL numbers for multi-phase flows with large density ratios

The discontinuity across interface of multi-phase flows with large density ratios usually poses great challenges for numerical simulations. The smoothed particle hydrodynamics (SPH) is a meshless method with inherent advantages in dealing with multi-phase flows without the necessity of tracking the moving interfaces. In this paper, we develop a new weakly-compressible SPH model for multi-phase flows with large density ratios while allowing large CFL numbers. In the present SPH model, the continuity equation is first modified by eliminating the influence from particles of different phases based on the simple fact that different phases will not contribute when calculating the density for immiscible multi-phase flows; thus, the modified continuity equation will only consider the influence from neighboring particles of the same phase. The pressure and density of the particles of other phases are then re-initialized by using the Shepard interpolation function. The present multi-phase SPH model has been tested by four numerical examples, including the two-phase hydrostatic water, standing waves, liquid sloshing, and dam breaking. It has been demonstrated that the present multi-phase SPH model can obtain satisfactory results stably, even at large CFL numbers, and this means that large time steps can be employed. Therefore, the present multi-phase SPH model can significantly save computational cost through using large time steps, especially for large-scale problems with a large number of particles.

Central-Moment Discrete Unified Gas-Kinetic Scheme for Incompressible Two-Phase Flows with Large Density Ratio

Central-Moment Discrete Unified Gas-Kinetic Scheme for Incompressible Two-Phase Flows with Large Density Ratio

Discrete Exterior Calculus Discretization of Two-Phase Incompressible Navier-Stokes Equations with a Conservative Phase Field Method

We present a discrete exterior calculus (DEC) based discretization scheme for incompressible two-phase flows. Our physically-compatible exterior calculus discretization of single phase flow is extended to simulate immiscible two-phase flows with discontinuous changes in fluid properties such as density and viscosity across the interface. The two-phase incompressible Navier-Stokes equations and conservative phase field equation for interface capturing are first transformed into the exterior calculus framework. The discrete counter part of these smooth equations is obtained by substituting with discrete differential forms and discrete exterior operators. We prove the boundedness of the method for the first order Euler forward and predictor-corrector time integration schemes in the DEC framework. With a proper choice of two free parameters, the scheme remains phase field bounded without the requirement of any ad hoc mass redistribution. We verify the scheme against several standard test cases (for interface capturing) comprising not only the flat domains but also the curved domains, leveraging the advantage that DEC operators are independent of the coordinate system. The results show excellent properties of boundedness, mass conservation and convergence. Moreover, we demonstrate the ability of the scheme towards handling large density and viscosity ratios as well as surface tension in the simulation of various two phase flow physical phenomena on flat or curved surfaces.

A mass conserving arbitrary Lagrangian–Eulerian formulation for three‐dimensional multiphase fluid flows

AbstractThe arbitrary Lagrangian–Eulerian (ALE) framework presented in [Sahin and Guventurk, An arbitrary Lagrangian–Eulerian framework with exact mass conservation for the numerical simulation of 2D rising bubble problem.Int J Numer Methods Eng. 2017;112:2110–2134] has been extended to simulate three‐dimensional immiscible incompressible multiphase fluid flows. The div‐stable side centered unstructured finite volume formulation is used for the discretization of the incompressible Navier–Stokes equations. A novel kinematic boundary condition that satisfies the local discrete geometric conservation law (DGCL) is implemented to ensure the mass conservation of each species at the machine precision. The pressure field is treated to be discontinuous across the interface with the discontinuous treatment of density and viscosity. Surface tension force is treated as a tangent force and discretized in a semi‐implicit form. Two different approaches for the computation of unit normal vector have been implemented: the least squares biquadratic surface fitting (LSBSF) and the mean weighted by sine and edge length reciprocals (MWSELR). The resulting large system of algebraic equations is solved in a fully coupled manner in order to improve the time step restrictions. As a preconditioner, an approximate matrix factorization similar to that of the projection method is employed and the parallel algebraic multigrid solver BoomerAMG provided by the HYPRE library, which is accessed through the PETSc library, has been utilized for the scaled discrete Laplacian of pressure and the diagonal blocks of mesh deformation equations. The accuracy of the proposed method is initially validated on the static bubble problem, since the surface tension force is highly sensitive to the accurate evaluation of the unit normal vector and the inaccuracies significantly contribute to unphysical velocities, called parasitic currents. The calculations indicate that the parasitic currents can be reduced to machine precision for the MWSELR method. Then, the proposed approach is applied to the single bubble rising in a viscous quiescent liquid for both low and high density ratios. The calculations produce accurate predictions of the bubble shape, center of mass, rise velocity, and so forth. Furthermore, the mass of each species is conserved at machine precision and discontinuous pressure field is obtained in order to avoid errors due to the incompressibility restriction in the vicinity of liquid‐liquid interfaces at large density and viscosity ratios.

Numerical investigation on the performance of gas-lift pump with large density ratio of liquid to gas

Heavy liquid metals such as sodium and liquid lead-bismuth eutectic (LBE) have excellent heat transfer capability and receive increasing attention as the primary coolant in fast reactors and accelerator driven systems (ADS). In ADS system, coolant system is designed as natural circulation, to simplify the primary system, improve the compactness of reactors and most important of all is to increase the inherent safety of reactor. Because the electromagnetic pump that drives the flow of liquid metal is eliminated in the coolant natural circulation system. In such situation, gas-lift technique named gas-lift pump could be adopted to enhance the coolant natural circulation capacity. The open research on the performance of gas-lift liquid metal pump is much less than that of air- lift water pump. The physical properties of large density ratio of liquid to gas, small kinematic viscosity and high surface tension of liquid metal is believed to have effects on the performance of gas-lift liquid metal pump. The present paper carries out some preliminary numerical simulation work on the two-phase flow characteristics of argon-liquid LBE in the up-riser tube and the performance of gas-lift liquid metal pump. Eulerian-Eulerian two-fluid model is used to describe the two-phase flow. Some interphase momentum transfer models from literature are used to account for the interfacial momentum transfer between liquid and gas phases. SST k-ω model with mixture treatment is used to compute the turbulence parameters of the homogeneous mixing phase in the present study. The influence of such parameters as argon injection superficial velocity, submergence ratio and diameter of up-riser tube on the discharged LBE mass flow rate and pressure drop of gas-lift pump are checked. The results could be helpful for the design of the gas-lift liquid metal pump.

Multiphase flow simulation with three-dimensional weighted-orthogonal multiple-relaxation-time pseudopotential lattice Boltzmann model

In this paper, based on two lattice models (D3Q19 and D3Q27), two three-dimensional weighted-orthogonal multiple-relaxation-time pseudopotential lattice Boltzmann (WMRT-PLB) models with tunable thermodynamic consistency and surface tension are developed in which the high-order terms of the equilibrium density distribution function and discrete forcing term in moment space are eliminated, and thus, the implementation of the collision process is simplified. The Chapman–Enskog analysis shows that the WMRT-PLB models can correctly recover the macroscopic Navier–Stokes equations in the low Mach number limit. Then, six classical multiphase flows benchmark cases are performed to validate the performance of the proposed model. The numerical results of the first three cases indicate that the developed WMRT-PLB models effectively weaken the non-physical coupling between kinetic viscosity and density, enhance the numerical stability because of the low spurious velocity, improve the computational efficiency by about 25% because of the simplification of the collision process, and increase the numerical accuracy in the dynamic problems. Meanwhile, the numerical results of the last three cases with the density ratio of 857.7 and the kinetic viscosity ratio of 1/15 agree well with the analytical solutions and experimental results reported in the literature. Note that it is also found that the simulation of droplet bouncing is still stable even when the Reynolds number is more than 3000, which shows the good numerical stability of the proposed model. It has the potential to be applied to the simulation of the complex multiphase flows with large density ratio and large Reynolds number.

High-fidelity simulation of a hydraulic jump around a surface-piercing hydrofoil

For a surface-piercing hydrofoil traveling at high speed, a turbulent hydraulic jump may arise at the intersection of the body with the free surface. This hydrodynamic phenomenon involves violent wave breaking, bringing great challenges for experimental analysis. In this work, a high-fidelity large eddy simulation is performed to study the turbulent air-entraining flow near foil. One advantage of the present simulation is that a quantitative analysis can be implemented even in the turbulent two-phase mixing region containing a large amount of entrained air, which is difficult for traditional experimental and theoretical approaches. We employ a conservative coupled level set/volume-of-fluid scheme to capture the free surface. A highly robust scheme is introduced to guarantee stability in simulating large density ratio two-phase flows. The present method is implemented based on a block-structured adaptive mesh, by which the efficiency of the high-fidelity simulation can be improved. The main flow features of the wedge-shaped hydraulic jump, including the wave patterns, free surface elevation, and frequency spectra, are compared with experimental data. We find that the flow structures show clear differences from those found in the canonical hydraulic jump, owing to the presence of the foil surface. Shoulder wave breaking starts at the trough of the mid-body, develops in a wedge shape, depends strongly on Froude number, and is responsible for most of the large-scale air entrainment. The properties of the turbulent hydraulic jump and some of the key quantities characterizing the air-entraining flow, including the spatial distribution of the bubble cloud, the void fraction, and the bubble/droplet size spectrum, are fully investigated for typical Froude numbers.

Simulating multi-phase sloshing flows with the SPH method

In this paper, the intense sloshing flow is simulated by Smoothed Particle Hydrodynamics (SPH) method successfully. First, the traditional SPH method is used for the simulation of sloshing process; and the obtained results for free surface shape and pressure evolution are in good agreement with the experimental data. Then, the δ-SPH method and particle shifting technique (PST) are introduced in the traditional SPH model to improve its stability and accuracy in simulating sloshing flow. Air phase does have an inescapable influence on water phase in intense sloshing process. Thus, an improved multiphase SPH model with pressure correction algorithm is utilized for interfacial sloshing simulation. The stability, accuracy and conservation of pressure correction algorithm are validated by the cases of hydrostatic test and dam breaking test. Besides, the comparison results show that unphysical pressure transmission at the interface between two fluids with large density ratio is effectively suppressed. Finally, the effect of air is considered in the multiphase flow simulation of sloshing process with presented multiphase δ-SPH and pressure correction utilized. The phenomenon that air phase is involved into inner water flow and then moves, rolls and rises in it is replicated accurately. It is found that both the shape of free surface and the air cavities during the slamming phase are well captured, which proves that the integrated multiphase SPH is effectively utilized for the simulation of such multiphase problem as sloshing.

Non-condensable gas bubble dissolution with a modified tunable surface tension multicomponent lattice Boltzmann model

A modified lattice Boltzmann pseudopotential multicomponent and multiphase model is developed and analyzed. The model is capable of simulating large density ratios and viscosity ratios while ensuring thermodynamic consistency, and has a tunable surface tension. An improved source term is introduced to achieve the tunable surface tension with little influence on the thermodynamic consistency. The effect of surface tension on the bubble dissolution process under pressure is investigated. The results show that dramatic deformation and oscillation amplitudes result from larger surface tension, leading to a smaller equilibrium radius, consistent with Laplace's law. The convection process on the interface is the dominant factor in the oscillation of the average dissolved gas concentration. The dissolved gas concentration takes longer to reach equilibrium than the bubble radius due to the slow diffusion process. The average dissolved gas concentration increases with increasing initial pressure difference and surface tension, and the relationship between the average gas phase partial pressure and the average dissolved gas concentration is linear for different surface tensions, consistent with Henry's law. Although surface tension influences the dissolved gas concentration, it does not impact Henry's coefficient. The average dissolved gas concentration and bubble radius display a linear relationship with the bubble energy.

Numerical simulation of multiphase flows using an enhanced Volume-of-Fluid (VOF) method

The main objective of the present work is to enhance the stability and accuracy of Volume-of-Fluid (VOF) method for simulating multiphase flow problems with large density ratios and moving interfaces. In order to accomplish this, a novel high-order TVD flux-limiter scheme is developed and applied to approximate the convective fluxes in the momentum, energy and volume fraction equations. Moreover, the classical implicit non-iterative PISO algorithm is modified according to SIMPLER algorithm and the combined model (PISOR) is then employed to handle the pressure-velocity coupling. Furthermore, the piecewise linear interface calculation (PLIC) based on the Efficient Least Squares Volume-of-Fluid Interface Reconstruction Algorithm (ELVIRA) is adopted for the treatment of the interface and curvature estimation. The robustness and consistency of the proposed modifications in handling violent free-surface flows involving interface rupture and coalescence are verified via simulation of several canonical test cases including: dam break over the dry and wet beds, rotating square patch of fluid and Rayleigh-Taylor Instability (RTI). The results show that, the proposed flux-limiter scheme can significantly suppress false-diffusion errors and maintain interface sharpness while preserving the boundedness and positivity of the volume fraction field. Moreover, it is found that the proposed PISOR model has robust convergence behavior in strongly coupled multiphysics problems and provides more stable and consistent results than the standard PISO and SIMPLER algorithms. The performance and efficiency of proposed models are further verified, for single phase flow, through analysis of transient entropy generation due to natural convection heat transfer in differentially heated cavity.